3 put(9)-11011740-1-11453830Some balance factors on insert path change.When a bf changes from 0 to +1 or –1 the subtree height increases and we need to go further up the tree adjusting balance factors.-160351920525

4 put(29)-11011740-11453830-1-260351205White node is the A node, nearest ancestor of newly inserted node whose bf becomes +2 or –2.RR imbalance => new node is in right subtree of right subtree of white node (node with bf = –2)-12529

6 Insert/PutFollowing insert/put, retrace path towards root and adjust balance factors as needed.Stop when you reach a node whose balance factor becomes 0, 2, or –2, or when you reach the root.The new tree is not an AVL tree only if you reach a node whose balance factor is either 2 or –2.In this case, we say the tree has become unbalanced.

7 A-NodeLet A be the nearest ancestor of the newly inserted node whose balance factor becomes +2 or –2 following the insert.Balance factor of nodes between new node and A is 0 before insertion.

8 Imbalance TypesRR … newly inserted node is in the right subtree of the right subtree of A.LL … left subtree of left subtree of A.RL… left subtree of right subtree of A.LR… right subtree of left subtree of A.

10 LR Rotation (case 1) Subtree height is unchanged.Before insertion.1ABABAfter insertion.CC2BA-1After rotation.B is a leaf prior to the insert.Subtree height is unchanged.No further adjustments to be done.

11 LR Rotation (case 2) Subtree height is unchanged.BLCRARhh-1C’LC1ABBLCRARhh-1CLCC2BA-1-1BLhC’LhCRh-1ARh1B is not a leaf prior to the insert, the insert takes place in the left subtree of C.Subtree height is unchanged.No further adjustments to be done.

17 New Balance Factor Of q Deletion from left subtree of q => bf--.Deletion from right subtree of q => bf++.New balance factor = 1 or –1 => no change in height of subtree rooted at q.New balance factor = 0 => height of subtree rooted at q has decreased by 1.New balance factor = 2 or –2 => tree is unbalanced at q.Initially q is the parent of the deleted node. At this time, one subtree of q is empty and the other is not empty. As q moves up towards the root, neither subtree of q is empty.

18 Imbalance ClassificationLet A be an ancestor (q) of the deleted node whose balance factor has become 2 or –2 following a deletion.Deletion from left subtree of A => type L.Deletion from right subtree of A => type R.Type R => new bf(A) = 2.So, old bf(A) = 1.So, A has a left child B.bf(B) = 0 => R0.bf(B) = 1 => R1.bf(B) = –1 => R-1.

21 R-1 Rotation New balance factor of A and B depends on b.BLCRA’Rh-1CLCb-121-1ABBLCRARh-1hbCLCCACRA’Rh-1BBLCLNew balance factor of A and B depends on b.Subtree height is reduced by 1.Must continue on path to root.Similar to LR.

22 Number Of Rebalancing RotationsAt most 1 for an insert.O(log n) for a delete.